Dear Andre, -
On 30 Oct 2016, at 19:39, Joyal, André <joyal.andre@uqam.ca> wrote:
Dear All,
The difference between a Grothendieck topos and an elementary topos is like the difference between a "frame"(= a locale) and a Heyting algebra.
That analogy brings out well the contrast between the infinitary, set-indexed structure (disjunctions, coproducts) and finitary algebraic (or essentially algebraic) structure that approximates it. Something missing, though, is the miracle by which an elementary topos S is also able to provide the information, extrinsic to the infinitary logic, of what a set might be in "set-indexed". Then frame in S, or bounded geometric morphism over S, substitute S for Set in our notions of frame or Grothendieck topology. With the elementary toposes, we can then play the trick where a geometric morphism has topological properties, with the codomain specifying the "sets" available and the domain specifying the space relative to it. As far as I know, we can't do anything similar with Heyting algebras. On the face of it, the miracle is associated with the subobject classifiers. However, that can't be the whole story. You see something similar, again in the passage from posets to categories, if you restrict to geometric constructions (colimits, finite limits). On the posets you might expect the finitary analogue of frames to be distributive lattices, but they are topologically not general enough. You only get spectral spaces, and not e.g. the real line. On categories, one candidate finitary analogue of Grothendieck toposes is arithmetic universes (= pretoposes with parametrized list objects), and again, like elementary toposes, they are able to supply infinities needed for spaces such as the real line. All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]